Worksheet: Combining Functions

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In this worksheet, we will practice adding, subtracting, multiplying, or dividing two given functions to create a new function and identifying the domain of the new function.

Q1:

What is the domain of the quotient 𝑓 𝑔 , in terms of the domains of 𝑓 and 𝑔 ? Assume that both domains are subsets of the set of real numbers.

  • A the union of the domain of 𝑓 and the domain of 𝑔
  • B the intersection of the domain of 𝑓 and the domain of 𝑔
  • C the larger of the domain of 𝑓 and the domain of 𝑔
  • D the intersection of the domain of 𝑓 and the domain of 1 𝑔
  • Ethe difference between the domain of 𝑓 and the domain of 𝑔

Q2:

Determine the common domain of the functions 𝑛 ( 𝑥 ) = − 7 𝑥 − 7 and 𝑛 ( 𝑥 ) = − 8 𝑥 − 6 4 .

  • A ℝ − { − 8 , − 7 , 8 }
  • B ℝ − { − 8 , 8 }
  • C ℝ − { 7 , 8 }
  • D ℝ − { − 8 , 7 , 8 }
  • E ℝ − { − 8 , − 7 }

Q3:

If 𝑓 and 𝑔 are two real functions where 𝑓 ( 𝑥 ) = 𝑥 − 1 𝑥 + 3 𝑥 − 4 and 𝑔 ( 𝑥 ) = 𝑥 + 3 , determine the value of ( 𝑓 + 𝑔 ) ( − 4 ) if possible.

  • A − 6 5
  • B − 1
  • C − 6
  • Dundefined

Q4:

Find the domain of the function 𝑓 ( 𝑥 ) = √ 𝑥 + 3 + √ 𝑥 − 7 .

  • A [ 3 , ∞ )
  • B ( − 3 , ∞ )
  • C [ 7 , ∞ )
  • D [ − 3 , ∞ )

Q5:

If 𝑓 ( 𝑥 ) = 𝑥 + 1 and 𝑔 ( 𝑥 ) = 𝑥 + 1 , then find and fully simplify an expression for ( 𝑓 ⋅ 𝑔 ) ( 𝑥 ) .

  • A 𝑥 + 𝑥 + 1
  • B 𝑥 + 𝑥 + 2
  • C 𝑥 + 𝑥 + 1
  • D 𝑥 + 𝑥 + 𝑥 + 1

Q6:

If 𝑓 ℝ → ℝ : where 𝑓 ( 𝑥 ) = 4 𝑥 − 4 , and 𝑔 [ − 8 , − 2 ) → ℝ : where 𝑔 ( 𝑥 ) = 5 𝑥 + 5 , find the value of ( 𝑓 + 𝑔 ) ( 5 ) if possible.

  • A16
  • B46
  • C36
  • Dundefined

Q7:

Given that 𝑓 ∶ ℝ → ℝ , where 𝑓 ( 𝑥 ) = 𝑥 − 1 9 , and 𝑔 ∶ [ − 2 , 1 3 ] → ℝ , where 𝑔 ( 𝑥 ) = 𝑥 − 6 , evaluate ( 𝑓 ⋅ 𝑔 ) ( 7 ) .

  • A − 2 4 0
  • B724
  • C − 7 7 4
  • D − 1 2

Q8:

If 𝑓 and 𝑔 are two real functions where 𝑓 ( 𝑥 ) = 𝑥 + 9 𝑥 + 1 5 𝑥 + 5 4 and 𝑔 ( 𝑥 ) = 𝑥 + 8 , determine the value of ( 𝑓 − 𝑔 ) ( − 6 ) if possible.

  • A − 5 3
  • B − 2
  • C1
  • Dundefined

Q9:

Given that , , and , determine in its simplest form.

  • A
  • B
  • C
  • D
  • E

Q10:

Given that 𝑛 ( 𝑥 ) = 5 𝑥 − 8 2 5 𝑥 − 4 ÷ 2 5 𝑥 − 3 0 𝑥 − 1 6 1 2 5 𝑥 + 8 , 𝑛 ( 𝑥 ) = 2 5 𝑥 − 4 5 0 𝑥 − 2 0 𝑥 + 8 , and 𝑛 ( 𝑥 ) = 𝑛 ( 𝑥 ) × 𝑛 ( 𝑥 ) , simplify the function 𝑛 , and determine its domain.

  • A 𝑛 = 2 , domain = ℝ − − 2 5 , 2 5 , 8 5
  • B 𝑛 = 1 2 , domain = ℝ − − 2 5 , 2 5
  • C 𝑛 = 2 , domain = ℝ − − 2 5 , 2 5
  • D 𝑛 = 1 2 , domain = ℝ − − 2 5 , 2 5 , 8 5
  • E 𝑛 = 5 2 , domain = ℝ − − 2 5 , 2 5 , 8 5

Q11:

Given that 𝑓 ∶ ( − ∞ , 4 ) → ℝ 𝑓 ( 𝑥 ) = 𝑥 + 5 s u c h t h a t and 𝑓 ∶ ( − 8 , 6 ) → ℝ 𝑓 ( 𝑥 ) = 2 𝑥 + 1 3 𝑥 + 1 5 , s u c h t h a t find 𝑓 𝑓 ( 𝑥 ) and state its domain.

  • A 2 𝑥 + 3 , 𝑥 ∈ ( − 8 , 6 )
  • B 2 𝑥 + 3 , 𝑥 ∈ ( − ∞ , 4 )
  • C 2 𝑥 + 3 , 𝑥 ∈ ( − 8 , 4 )
  • D 2 𝑥 + 3 , 𝑥 ∈ ( − 8 , 4 ) − { − 5 }
  • E 2 𝑥 + 3 , 𝑥 ∈ ( − ∞ , 4 ) − { − 5 }

Q12:

Given that 𝑓 ∶ ℝ → ℝ 𝑓 ( 𝑥 ) = − 3 𝑥 − 4 w h e r e and 𝑔 ∶ ( 1 , 7 ) → ℝ 𝑔 ( 𝑥 ) = − 2 𝑥 − 4 , w h e r e find the value of 𝑓 𝑔 ( − 1 ) if possible.

  • A − 1
  • B 1 2
  • C0
  • Dnot defined

Q13:

Given that 𝑓 ∶ ( − ∞ , 2 ] ⟶ ℝ 𝑓 ( 𝑥 ) = 𝑥 + 5 s u c h t h a t and 𝑓 ∶ ( − ∞ , − 1 ) ⟶ ℝ 𝑓 ( 𝑥 ) = 2 𝑥 − 𝑥 − 6 , s u c h t h a t find 𝑓 𝑓 ( 𝑥 ) and state its domain.

  • A 2 𝑥 − 𝑥 − 6 𝑥 + 5 , 𝑥 ∈ ( − ∞ , − 1 ]
  • B 2 𝑥 − 𝑥 − 6 𝑥 + 5 , 𝑥 ∈ ( − ∞ , − 1 )
  • C 2 𝑥 − 𝑥 − 6 𝑥 + 5 , 𝑥 ∈ ( − ∞ , 2 ]
  • D 2 𝑥 − 𝑥 − 6 𝑥 + 5 , 𝑥 ∈ ( − ∞ , − 1 ) − { − 5 }
  • E 𝑥 + 5 2 𝑥 − 𝑥 − 6 , 𝑥 ∈ ( − ∞ , − 1 ) − { − 5 }

Q14:

If 𝑓 and 𝑔 are two real functions where 𝑓 ( 𝑥 ) = 2 𝑥 + 2 𝑥 < − 3 , 𝑥 − 4 − 3 ≤ 𝑥 < 0 , i f i f and 𝑔 ( 𝑥 ) = 5 𝑥 determine the domain of the function 𝑔 𝑓 .

  • A [ − 3 , 0 )
  • B ( − ∞ , − 3 )
  • C ℝ − { 0 }
  • D ( − ∞ , 0 )

Q15:

If 𝑓 and 𝑔 are two real functions where 𝑓 ( 𝑥 ) = 𝑥 − 5 𝑥 and 𝑔 ( 𝑥 ) = √ 𝑥 + 1 , find the domain of the function ( 𝑓 + 𝑔 ) .

  • A [ − 1 , ∞ ) − { 0 , 5 }
  • B ( − ∞ , − 1 ]
  • C ℝ − { 0 , 5 }
  • D [ − 1 , ∞ )
  • E [ 1 , ∞ )

Q16:

If 𝑓 ∶ ( − 7 , 8 ] → ℝ where 𝑓 ( 𝑥 ) = 𝑥 − 2 , and 𝑓 ∶ [ − 8 , 4 ] → ℝ where 𝑓 ( 𝑥 ) = 4 𝑥 + 8 𝑥 + 3 , find ( 𝑓 − 𝑓 ) ( 𝑥 ) and the domain of ( 𝑓 − 𝑓 ) .

  • A 4 𝑥 + 7 𝑥 + 5 , 𝑥 ∈ ( − 7 , 8 ]
  • B 4 𝑥 + 7 𝑥 + 5 , 𝑥 ∈ [ − 8 , 4 ]
  • C 4 𝑥 + 7 𝑥 + 5 , 𝑥 ∈ [ − 7 , 4 )
  • D 4 𝑥 + 7 𝑥 + 5 , 𝑥 ∈ ( − 7 , 4 ]
  • E − 4 𝑥 − 7 𝑥 − 5 , 𝑥 ∈ ( − 7 , 4 ]

Q17:

If 𝑓 ∶ ℝ ⟶ ℝ where 𝑓 ( 𝑥 ) = 𝑥 − 1 7 , and 𝑔 ∶ [ − 2 5 , 4 ] ⟶ ℝ where 𝑔 ( 𝑥 ) = 𝑥 − 1 1 , then find ( 𝑓 + 𝑔 ) ( 𝑥 ) and its domain.

  • A ( 𝑓 + 𝑔 ) ( 𝑥 ) = 2 𝑥 − 1 1 , ( 0 , 4 ]
  • B ( 𝑓 + 𝑔 ) ( 𝑥 ) = 2 𝑥 − 1 7 , [ 0 , 4 ]
  • C ( 𝑓 + 𝑔 ) ( 𝑥 ) = 2 𝑥 − 2 8 , [ 0 , 4 ]
  • D ( 𝑓 + 𝑔 ) ( 𝑥 ) = 2 𝑥 − 2 8 , ( 0 , 4 ]

Q18:

If 𝑓 ℝ → ℝ : where 𝑓 ( 𝑥 ) = 4 𝑥 + 4 , and 𝑓 ( − 9 , 6 ] → ℝ : where 𝑓 ( 𝑥 ) = 𝑥 − 1 , find and fully simplify ( 𝑓 − 𝑓 ) ( 𝑥 ) and determine the domain of ( 𝑓 − 𝑓 ) .

  • A 3 𝑥 + 5 , 𝑥 ∈ ℝ
  • B 3 𝑥 + 5 , 𝑥 ∈ ( − 9 , 6 ]
  • C 3 𝑥 + 5 , 𝑥 ∈ [ − 9 , 0 ]
  • D 3 𝑥 + 5 , 𝑥 ∈ ( − 9 , 0 )
  • E 3 𝑥 + 5 , 𝑥 ∈ ( − ∞ , 6 ]

Q19:

If 𝑓 ℝ → ℝ : where 𝑓 ( 𝑥 ) = − 𝑥 − 1 , and 𝑓 ( − 9 , 1 ) → ℝ : where 𝑓 ( 𝑥 ) = 5 𝑥 − 3 , find ( 𝑓 + 𝑓 ) ( 𝑥 ) and the domain of ( 𝑓 + 𝑓 ) .

  • A 4 𝑥 − 4 , 𝑥 ∈ ℝ
  • B 4 𝑥 − 4 , 𝑥 ∈ ( − 9 , 1 )
  • C 4 𝑥 − 4 , 𝑥 ∈ [ − 9 , 0 ]
  • D 4 𝑥 − 4 , 𝑥 ∈ ( − 9 , 0 )
  • E 4 𝑥 − 4 , 𝑥 ∈ ( − ∞ , 1 )

Q20:

If 𝑓 and 𝑔 are two real functions where 𝑓 ( 𝑥 ) = 𝑥 − 5 𝑥 and 𝑔 ( 𝑥 ) = √ 𝑥 + 4 , determine the domain of the function ( 𝑓 ⋅ 𝑔 ) .

  • A [ − 4 , ∞ ) − { 0 , 5 }
  • B ( − ∞ , − 4 ]
  • C ℝ − { 0 , 5 }
  • D [ − 4 , ∞ )
  • E [ 4 , ∞ )

Q21:

Given that and find ( 𝑓 ⋅ 𝑓 ) ( 𝑥 ) 1 2 and state its domain.

  • A 5 𝑥 − 2 2 𝑥 + 8 2 , 𝑥 ∈ ℝ +
  • B 5 𝑥 − 2 2 𝑥 + 8 2 , 𝑥 ∈ ] − 9 , 1 ]
  • C 5 𝑥 − 2 2 𝑥 + 8 2 , 𝑥 ∈ [ 0 , 1 [
  • D 5 𝑥 − 2 2 𝑥 + 8 2 , 𝑥 ∈ ] 0 , 1 ]
  • E 5 𝑥 − 2 2 𝑥 + 8 2 , 𝑥 ∈ ] − 9 , ∞ [

Q22:

Given that 𝑓 ∶ ( 3 , 6 ] → ℝ 𝑓 ( 𝑥 ) = 𝑥 + 1 0 𝑥 + 2 5 s u c h t h a t and 𝑓 ∶ ( − 1 , 9 ) → ℝ 𝑓 ( 𝑥 ) = 𝑥 − 3 , s u c h t h a t find ( 𝑓 ⋅ 𝑓 ) ( 𝑥 ) and state its domain.

  • A 𝑥 + 7 𝑥 − 5 𝑥 − 7 5 , 𝑥 ∈ [ 3 , 6 )
  • B 𝑥 + 7 𝑥 − 5 𝑥 − 7 5 , 𝑥 ∈ ( − 1 , 9 )
  • C 𝑥 − 3 0 𝑥 + 1 0 𝑥 − 7 5 , 𝑥 ∈ ( 3 , 6 ]
  • D 𝑥 + 7 𝑥 − 5 𝑥 − 7 5 , 𝑥 ∈ ( 3 , 6 ]

Q23:

If 𝑓 and 𝑔 are two real functions where 𝑓 ( 𝑥 ) = 𝑥 + 5 0 < 𝑥 < 2 , 2 𝑥 + 5 𝑥 ≥ 2 , i f i f and 𝑔 ( 𝑥 ) = 𝑥 , find the domain of the function ( 𝑓 ⋅ 𝑔 ) .

  • A [ 2 , ∞ )
  • B ( 0 , 2 )
  • C ( 0 , ∞ ) − { 2 }
  • D ( 0 , ∞ )
  • E ℝ

Q24:

Given that 𝑓 and 𝑔 are two real functions where 𝑓 ( 𝑥 ) = 𝑥 + 2 𝑥 and 𝑔 ( 𝑥 ) = √ 5 − 𝑥 , find the value of 𝑓 𝑔 ( 5 ) if possible.

  • A0
  • B − 3 5
  • C35
  • Dundefined

Q25:

Given that 𝑓 and 𝑔 are two real functions where 𝑓 ( 𝑥 ) = 𝑥 − 1 and 𝑔 ( 𝑥 ) = √ 𝑥 + 5 , find the value of 𝑔 𝑓 ( − 2 ) if possible.

  • A − √ 3 3
  • Bundefined
  • C3
  • D √ 3 3
  • E √ 3

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